Four-Qubit Entanglement Classification from String Theory

Abstract

We invoke the black hole/qubit correspondence to derive the classification of four-qubit entanglement. The U-duality orbits resulting from timelike reduction of string theory from D = 4 to D = 3 yield 31 entanglement families, which reduce to nine up to permutation of the four qubits.PACS numbers: 11.25.Mj, 03.65.Ud, 04.70.Dy Keywords: black hole, U-duality, qubit, entanglement Recent work has established some intriguing correspondences between two very different areas of theoretical physics: the entanglement of qubits in quantum information theory (QIT) and black holes in string theory. See [1] for a review. In particular, there is a one-to-one correspondence between the classification of three qubit entanglement [2] and the classification of extremal black holes in the ST U supergravity theory [3,4] that appears in the compactification of string theory from D = 10 to D = 4 dimensions. Moreover, the Bekenstein-Hawking black hole entropy is provided by the three-way entanglement measure.The purpose of this paper is to use this black hole/qubit correspondence to address the much more difficult problem of classifying four-qubit entanglement, currently an active area of research in QIT as experimentalists now control entanglement with four qubits [5]. Although two and three qubit entanglement is wellunderstood, the literature on four qubits can be confusing and seemingly contradictory, as illustrated in Table I. This is due in part to genuine calculational disagreements, but in part to the use of distinct (but in principle consistent and complementary) perspectives on the criteria for classification. On the one hand there is the "covariant" approach which distinguishes the orbits of the equivalence group of Stochastic Local Operations and Classical Communication (SLOCC) by the vanishing or not of covariants/invariants. This philosophy is adopted for the three-qubit case in [2,13], for example, where it was shown that three qubits can be tripartite entangled in two inequivalent ways, denoted W and GHZ (Greenberger-Horne-Zeilinger). The analogous four-qubit case was treated, with partial results, in [14]. On the other hand, there is the "normal form" approach which considers "families" of orbits. Any given state may be transformed into a unique normal form. If the normal form depends on some of the algebraically independent SLOCC invariants it constitutes a family of orbits parametrized by these invariants. On the other hand a parameter-independent family contains a single orbit. This philosophy is adopted for the four-qubit case |Ψ = a ABCD |ABCD A, B, C, D = 0, 1 in [11,12]. Up to permutation of the four qubits, these authors found 6 parameter-dependent families calledFor example, a family of orbits parametrized by all four of the algebraically independent SLOCC invariants is given by the normal form G abcd :(1)To illustrate the difference between these two approaches, consider the separable EPR-EPR state (|00 + |11 ) ⊗ (|00 + |11 ). Since this is obtained by setting b = c = d = 0 in (1) it belongs to the G abcd fa...

The black-hole/qubit correspondence: an up-to-date review

“…It follows that there are 31 nilpotent orbits for four qubits under SLOCC [22]. For each nilpotent orbit there is precisely one family of SLOCC orbits since each family contains one nilpotent orbit on setting all invariants to zero.…”

“…Another useful aspect of this correspondence is that the classification problem of certain types of black hole can be mapped to the classification problem of entanglement types of qubit systems [2,3,[21][22][23] .…”

Localized quadripartite entanglement

“…In addition, a good entanglement measure can not only distinguish the given entangled state from others (in usual, the separable states), but also should be an entanglement monotone which is not increased under stochastic local operations and classical communications(SLOCC) [10]. Therefore, although many authors have classified the multipartite quantum states into various inequivalent classes, they can not provide a good and direct entanglement measure for each class of quantum states [11][12][13][14][15][16][17][18][19][20]. In particular, in order to measure the * quaninformation@sina.com; ycs@dlut.edu.cn GHZ type entanglement of multipartite quantum states, some authors defined an n-tangle for multipartite quantum pure states to generalize the 3-tangle [20].…”

Peccei–Quinn Transformations and Black Holes: Orbit Transmutations and Entanglement Generation

“…Important achievements have been reached in this context, and some others are underway; just to name a few, the EBH entropy in the so-called N = 2 supergravity STU model [38,39] was associated with tripartite entanglement measurement and classification in QIT [26,40], and this was further extended to the hitherto unsolved issue of the classification of four qubits entanglement [41]. Moreover, the Hilbert space for qubits was traced back to wrapped branes inside the cohomology of the extra dimensions [32,42], and exceptional groups E 7 and E 6 were respectively related to the tripartite entanglement of seven qubits [43] and to the bipartite entanglement of three qutrits [44].…”